The integral provided in the image is:

$\int \frac{x^5 \, dx}{\sqrt{4 + x^5}}$

To solve this, we can apply substitution methods. Let's proceed step by step:

Step 1: Substitution

Let: $u = 4 + x^5$ Then, the derivative of $u$ is: $du = 5x^4 \, dx$ or: $\frac{du}{5} = x^4 \, dx$

We also have: $x^5 = u - 4$

Step 2: Substituting in the Integral

Now rewrite the integral in terms of $u$: $\int \frac{x^5 \, dx}{\sqrt{4 + x^5}} = \int \frac{(u - 4) \cdot \frac{du}{5}}{\sqrt{u}}$

This simplifies to: $\frac{1}{5} \int \frac{u - 4}{\sqrt{u}} \, du$ We can now split the terms: $\frac{1}{5} \int \frac{u}{\sqrt{u}} \, du - \frac{4}{5} \int \frac{1}{\sqrt{u}} \, du$ which simplifies further: $\frac{1}{5} \int u^{1/2} \, du - \frac{4}{5} \int u^{-1/2} \, du$

Step 3: Solving the Integrals

The first integral: $\int u^{1/2} \, du = \frac{2}{3} u^{3/2}$ The second integral: $\int u^{-1/2} \, du = 2u^{1/2}$

Step 4: Final Expression

Substitute these results back into the equation: $\frac{1}{5} \left( \frac{2}{3} u^{3/2} \right) - \frac{4}{5} \left( 2 u^{1/2} \right)$ Simplifying this: $\frac{2}{15} u^{3/2} - \frac{8}{5} u^{1/2}$

Finally, substitute $u = 4 + x^5$ back: $\frac{2}{15} (4 + x^5)^{3/2} - \frac{8}{5} (4 + x^5)^{1/2} + C$

Thus, the solution to the integral is: $\frac{2}{15} (4 + x^5)^{3/2} - \frac{8}{5} (4 + x^5)^{1/2} + C$

Let me know if you would like further clarification on any steps or more details.

Related Questions:

1. How can we apply substitution to solve different types of integrals?
2. What are the common integral techniques involving square roots?
3. How does the choice of substitution affect the complexity of solving an integral?
4. What other methods could be used to solve integrals like this one?
5. What is the importance of constant $C$ in indefinite integrals?

Tip: When solving integrals, always look for patterns or expressions inside the integral that resemble a function and its derivative to choose an effective substitution.